# How to solve optimization problem

I am a newbie in Optimization. I have this Optimization problem, could anyone help how can I analyze it and solve it?

\begin{align}\label{Problem_formulation} \mathbb P_1& ~~~~~~\mathop{\max}_{{ \eta, x_0 }} ~~~~{B}\log _2 \left(1+ \frac{{{P_M}\Vert {\textbf g}_d\Vert ^2}{( {\frac{c}{{4\pi {q}}}} )^2 {x_3}^{ - 2}{e^{ - K{x_3}}}}} {{{\sigma_w^2}}}\right)+\sum\nolimits_{f = 1}^{[S/\eta ]} \frac{{{f^{ - \delta }}}}{{\sum\nolimits_{f = 1}^{[S/\eta ]} {{f^{ - \delta }}} }}~~{B}\log _2 \left(1+ \frac{{{P_S}\Vert {\textbf h}_1\Vert ^2}{( {\frac{c}{{4\pi {q}}}} )^2 {x_0}^{ - 2}{e^{ - K{x_0}}}}} {{{\sigma_w^2}}}\right)\nonumber \\ &\mathop{\rm{s.t.}} \;\;~\eta \in [0,1], ~~ \forall f \in F \nonumber\\ &\;\;\qquad \sum\nolimits_{f = 1}^{[S/\eta ]} {b_f} \leq S \end{align}

### System parameters

• System bandwidth $$B = 10^{6}$$
• Number of files $$F = 10^{4}$$
• SBS cash capacity $$S = 100$$
• MBS-user distance $$x_{3} = 15$$
• Noise power in Watts $$\sigma = 10^{-90/10}$$
• Light speed $$c = 3 \cdot 10^{8}$$
• Operating frequency $$q = 10^{12}$$
• Molecular absorption coefficient $$K = 0.0016$$
• Path loss exponent for NLOS direct link MBS-user $$\alpha_{N} = 4$$
• Path loss exponent for LOS link SBS-user $$\alpha_{L} = 2$$
• Transmit power MBS in Watts $$P_{M} = 10^{30/10}$$
• Transmit power SBS in Watts $$P_{S} = 10^{30/10}$$
• Direct link MBS-user gain $$G_{gd} = 4$$
• LOS link SBS-user gain $$G_{h} = 4$$
• Skewness factor $$\delta = 1$$

I plot the function to see its curve:

• You will get better answers at or.stackexchange.com Apr 27 at 23:37
– Mark
May 9 at 7:06

If you have access to a non linear constrained solver (In MATLAB you may look at fmincon()) just use it.
If you're limited to uncosntrained solver (Like MATLAB's fminunc()). The method I'd start with is writing the Lagrangian form of the problem and use the Penalty Method.